Velocity and acceleration algorithm

[cupo]

Homework Statement

Why are (2) and (4) equations more accurately than (1) and (3) ? Why is 7*dt in (4) equation ? What kind of equations are (2) and (4) ? What method they used to write (3) and (4) equations?

Homework Equations

Velocity:
(1) v = (x - x[i-1]) / (t - t[i-1])
(2) v = (x[i+1] - x[i-1]) / (2*dt)

Acceleration:
(3) a = (v - v[i-1]) / (t - t[i-1])
(4) a = (2*x[i+2] - x[i+1] - 2*x - x[i-1] + 2*x[i-2]) / (7*dt)

The Attempt at a Solution

Verlet algorithm
Finite difference

 

[haruspex]

The problem with (1) and (3) is that there's a time shift between the input positions and the computed velocities and accelerations. E.g (1) is really estimating the velocity at step i-0.5. (2) corrects that.
(4) achieves the same result, but I'm not sure why it's quite as it is. (Shouldn't it have dt² at the end?) If you start with the 'smoothed' acceleration expression (2*a_i+1+3*a_i+2*a_i-1)/7 and then substitute for those a_i using the forms a_i.Δt = v_i+.5-v_i-.5 and v_i.Δt = x_i+.5-x_i-.5 you arrive at (4) (with dt²). But why start with (2*a_i+1+3*a_i+2*a_i-1)/7 rather than e.g. (a_i+1+2*a_i+a_i-1)/4 I don't know.